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Functional equations related to inner product spaces. (English) Zbl 1177.39041

Summary: Let \(V,W\) be real vector spaces. It is shown that an odd mapping \(f:V\to W\) satisfies
\[ \sum^{2n}_{i=1}f(x_i-1/2n\sum_{j=1}^{2n}x_j)=\sum_{i=1}^{2n}f(x_i)-2_nf(1/2n\sum_{i=1}^{2n}x_i) \]
for all \(x_1,\dots,x_{2n}\in V\) if and only if the odd mapping \(f:V\to W\) is Cauchy additive. Furthermore, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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References:

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